Jennie and Craig… (Relativity)

Jennie and Craig travel through space in two identical spaceships along the same straight-line path but at different speeds. Each ship has a rest mass of $4.5*10^6 kg$. Craig informs Jennie that he has made length measurements of his ship and finds it to be 150 m long. He also finds thatJennie’s is 120 m long. A)From Jennie’s point of view, how long is Craig’s ship? B)What is the speed of Craig’s ship relative to Jennie’s? Now suppose both ships approach Earth from opposite directions, each with a speed of 0.6c relative to Earth C)What is the magnitude of either ship’s relativistic momentum with respect to the Earth? D) What is either ship’s total relativistic energy with respect to the Earth? E) What is the speed of one ship relative to the other?

B) Both ships are identical. Hence they are the same length in the same reference system. In two reference systems S’ moving at speed V relative to S the longitudinal length contracts in S’ as

$L = L0*\sqrt{(1-V^2/C^2)}$, where $L0$ is the longitudinal length in S

$1-V^2/C^2 =(L/L0)^2$

$V^2/C^2 =1-(L/L0)^2$

$V= c*\sqrt{(1-(L/L0)^2)} =3*10^8*\sqrt{(1-(120/150)^2)} =1.8*10^8 m/s =0.6*C$

A) From Jennie point of view the Craig ship is 120 m, the ships are identical hence they contract the same.

C) The relativistic momentum is defined as

$P_r = P/\sqrt{(1-V^2/C^2)}$

where P is the non-relativistic momentum $P =M0*V$ ($M0$ is the mass in the system S )

Hence $Pr = M0*V/\sqrt{(1-V^2/C^2)} =4.5*10^6*0.6*3*10^8/(\sqrt{(1-0.6^2)} =1.0125*10^{15} kg*m/s$

D) The relativistic energy is defined as

$E =M*C^2$ where M is the mass in the S’ system (moving system)

$E = M0*C^2/\sqrt{(1-V^2/C^2)} =4.5*10^6*(3*10^8)^2/\sqrt{(1-0.6^2)} =5.0625*10^{23} J$

E) The addition of speeds is

$s= (U+V)/(1+(U*V/C^2))$

where $U=V =0.6*C$ here, hence

$S =(0.6+0.6)*C/(1+0.6*0.6/1) =0.88*C =2.64*10^8 m/s$